I got an email from a fellow edu-blogger a couple of days ago asking for my input on the subject of academic rigor. Specifically this person asked:

Is the quest for more rigor an issue for you? Is it good, bad, meaningless? What does rigorous teaching look like in your classroom?

I hope she doesn’t mind my sharing the answer, because after writing it I thought it’d make a good blog post. I said:

For me, “rigor” in the context of intellectual work refers to thoroughness, carefulness, and right understanding of the material being learned. Rigor is to academic work what careful practice and nuanced performance is to musical performance, and what intense and committed play is to athletic performance. When we talk about a “rigorous course” in something, it’s a course that examines details, insists on diligent and scrupulous study and performance, and doesn’t settle for a mild or informal contact with the key ideas.

Example: A rigorous course in geometry goes beyond just memorization of formulas, applications to simple geometric exercises, and “hand-waving” attempts at proof. Instead, such a course treats details as important, the ability to explain on a deep level the truth of formulas and results as a key goal for students, and sets a high bar for the exactitude of mathematical arguments. Euclid’s “Elements” for example is the prototype of the rigorous treatment of geometry. It’s not a difficult work to understand, necessarily; in fact one of the enduring qualities of the Elements is the clarity and precision of not only each individual proposition but also in how the overall collection of propositions fits together. By contrast, many modern books on geometry are highly non-rigorous, omitting details, putting theorems out of order, and defining a proof as a “reasonable explanation” only.

Is rigor good? It depends on the audience and the goals of the class. When I teach a geometry course for junior and senior Math Education majors, rigor is of the utmost importance because I want those pre-service teachers to go into their classrooms with tough, precise minds for the sake of their students. If I were to teach a geometry class for fifth-graders, on the other hand, I think rigor would obscure the subject, and I would depend a lot more on intuitive explanations and perhaps constructivist techniques for discovering key ideas in geometry and save rigorous proofs for another day. Similarly, when I teach calculus at my college, the audience is about 50% business majors, and so we designed the course not to cover much theory. This is not a rigorous treatment of calculus, but it is more effective for the students than if we included the epsilon-delta proofs and what not.

The quest for more rigor is most important in the post-calculus courses I teach (geometry, abstract algebra, and introduction to proof). These are subject areas where precision and detail-orientation are essential for a complete understanding of the material. Students are not allowed to give examples when a proof is called for, and I nitpick every little thing in their proofs up to and including the choice of punctuation and prepositions. [If any of the five who took this course from me this past semester are reading this, feel free to chime in with an “Amen.” – RT.] At the calculus level and below, I lay off on the theory but the rigor in the course comes from getting details of mechanical calculations right. And this is a big issue, because students in high school are generally taught only to produce a correct answer, not a clear and detailed solution. I am on a mission to make sure students can not only get right answers but also communicate their methods in a clear and audience-appropriate way, and that’s what “rigor” looks like there.

[After-the-fact note: To clarify, in calculus I insist on details in mechanical calculations but also on the details of processes and in paying attention to nuances in solving application-style problems. For example, students know that if you just set and solve for , that this doesn’t give you an inflection point; and in an optimization problem you can’t just find the critical number of the model function, you must also test it with the First or Second Derivative Test to see if it really yields a maximum. Or at least, they don’t complain when they forget to do it and I take off points!]

I have two kids, ages 3 and 5. (There’s a third one on the way in three weeks, but that’s another story!) I’m pretty rigorous with them, too — when the 5-year old says “Mimi comed to our house this weekend” I correct her grammar, and she gets it right the next time. You have to do it in a gentle way, but getting details right now will help them get the more complicated things right later. If I were to project myself out of higher ed and into the K-12 sphere I could see my teaching being “rigorous” in that kind of way — insisting that kids get the details right and not gloss over things, but doing it in a lovingly persistent way. I wish more K-12 teachers would do this, though, because it’s obvious from my freshmen in the last 4-5 years that this isn’t happening (or at least it’s not sticking).

[Final note: That last sentence isn’t a slam on either my freshmen, who were really quite excellent this year in calculus, or their teachers. It’s an observation, and I stand by it. I can show you their work at the beginning of the semester if you don’t believe me. Why this kind of “rigor” is not sticking with them is something I can’t fully explain because I don’t know what was going on with them in high school. Is it them? Is it their teachers? Is it the system? Is it the preponderance of standardized testing, which makes rigor more or less irrelevant? Comment!]

3 responses to “What does academic rigor look like?”

It’s the system — kids are generally used to learning and flushing it away after the test. Knowledge isn’t carried forward much, esp. if they think “i’ll never need this again” or “they’ll just reteach what I need in my next class.”

KC — I think its too easy to blame “the system” — it is up to teachers to buck the system if it gets in the way of educating students.

One simple example: To prevent “flushing after the test”, I have made all tests cumulative, by including some material from previous chapters. In fact, students know that if most of the class missed a particular concept, they will see the concept again on the next test — and I will not take class time to reteach it (aside from covering the concept when reviewing the test).

When considering Robert’s observation of incoming freshmen, I would venture to guess that too many teachers have burned out and have largely given up with the “fight” for rigor…they get tired of pushing against a general student drive for low expectations…they get disillusioned when forced to prepare students for low level standardized tests…students do extremely well on these and then parents complain about the low grades on teacher generated exams, because they don’t understand that the teacher has higher expectations for rigor than what is demanded on those standardized tests. It becomes easy to fall into a friendly relationship with students, demanding enough to make sure they perform on the end of course state test, giving high grades along the way — students are happy, parents are happy, administrators are happy and rigor is lost.

It takes some consistent, loving, encouragement — some charisma– some marketing of a sort — to get students to by into the focus and mental effort required to fully engage with academic material. It is a conscious fight every day — a fight which must be fought with enthusiasm and, at least in my case, must be supported by prayer and regular personal demonstrations of academic rigor in the classroom — I am always looking for new and interesting problems to present or new applications of the material or ways to tie the material to other subjects of interest to a particular class or student…students need to see rigor modeled. I try to do this by stretching myself and presenting these stretches in class. [yes, this means, I occasionally present material which will not be tested in a formal way]

Perhaps an important question in this discussion is this: Who, in “the system” is supporting K-12 teachers in a way that encourages them in this fight?